hello welcome back I'm Jason with math and science comm today the title of this lesson is called logarithm change of base formula and also solving more exponential and logarithm equations I told you at the very beginning the most important thing about logarithms that we're going to use them for at least in the near term is in solving equations and specifically solving exponential equations and we have solved some exponential equations and we have used logarithms but by now you probably figured out that some equations become difficult to solve if the logarithms or to get to get a numerical answer for if the logarithms that you're dealing with are in a crazy weird basis like for instance we've we've talked about the fact that you can you have logarithms in a base two you can have base five logarithms base three logarithms so you have those exact answers if you can take logarithms with all these different bases but a lot of times what happens is you have a calculator in your hand and you want a number but the calculator doesn't have usually a logarithm with a base five and a base six and a base 10 it makes eleven they usually don't have all of those things so how do we evaluate and solve equations to get numerical answers when the only button on the calculator we have is the regular logarithm button how do we do that so it turns out that there is a what we call a logarithm change of base formula and that is a really neat little tool to have in your back pocket because it allows you to take logarithms of any base you want by changing the way that the problem looks and it's a little bit complicated to say in words so I'm gonna write it down for you before we get to the actual change of base formula I want you to kind of just pull your calculator out whatever a little calculator you have generally there's going to be two logarithm buttons on the calculator okay and the first one will be called l OG if you just see it an L og a log button it means it's a base 10 logarithm you don't have to put the number 10 there a little base 10 there because you know if it says L og its base 10 so L og on your calculator means base 10 right and then you have another button that's called L in and this thing's called base it's called a natural logarithm and this is the base e logarithm where study base in natural logarithms later I don't want to get into it now but just suffice to say there's a very special base with a labeled with the letter e and that is just a number E is a rational number like pi or like square root of 2 e is two point seven one and then infinite non repeating decimals after we're gonna get all into Y E is important a little bit later so forget about it for now but I want to just let you know that your calculator is only gonna have two buttons base ten logarithm and in base e logarithm right now computer if you go to a computer you can take any base logarithm you want but on a handheld calculator these are the only ones you'll probably see so how do we do something for instance like if I want to figure out you know what is the logarithm base for of the number 17 how do I do that like how do I even type it into the calculator there is no way to do usually logarithms of different bases in a calculator so we have a formula that I'm gonna write on the board that's in all of the you know algebra books it's also in the calculus books because we use it quite a bit in calculus and it's called the change of base formula so let me write it down and explain how it basically works all right so this is the meat potatoes of this lesson this is called the log change of base formula and you're gonna find out that it allows us to do things like this actually really really easily alright and here's what it is it's not very hard or lengthy to write down here's what it is logarithm base a of the number X any base I want can be written as the logarithm of any other base I want of the number X same as this divided by logarithm same base B of a now I'm fully aware that most of you that have never seen this before are going to look at this and say this makes no sense to me so here in just a minute it will make a lot of sense I promise you right but what it allows you to do is it allows you to take logarithms of crazy weird bases only by using well it's used for more than this but mostly you're using it to be able to do it on a calculator because you don't have bay fifteen logarithms or base seven logarithm buttons on your calculator I am NOT going to prove this right now here in the beginning of the lesson what I want to do is show you how to use it and then if you stick with me to be very into the lesson I'm gonna drive and show you where exactly this thing comes from so just stick around if you want to know where it comes from now what this is saying in a nutshell it's easier to see with a simple little example what if I wanted to take the logarithm write the base three logarithm of the number nine now I just told you that the base three logarithm button is not in your calculator so there's no way to put it in there so you can see log base a of the number X this is log in this case base a isn't number three and the number I'm taking the log is here so this is basically this what this is saying is I can write this logarithm as follows I can change the base of this logarithm from base a to any other base I want notice B is the base on the right and a is the base on the left so if I want to I can change this thing to a logarithm base 2 of the number nine divided by logarithm base 2 of the number three I want to let that sink in for a second what it's saying is I'm trying to take base 3 log of the number nine I can instead change that to the logarithm of the number nine what you're taking with original you're taking the log of over here you're taking the log over here I want to make it a base two for whatever reason so I have to take base two logarithm on mine but then I have to divide it by the base two log of what the original bases see the base is three right here okay now this isn't so helpful because I don't have a base two logarithm on my calculator either I don't have a base two button on my calculator but mathematically a base two log of 9 divided by a base two log of three is equal to this right here now you can you can make any base up on the right-hand side that you want for instance it doesn't have to be base - I can make this log base 7 of the number nine as long as I divided by log base 7 of the number three you see what I've done I'm still taking the log of nine I'm still taking the log of the base but now I've changed it to a log base seven now the real reason that we use this thing so much is for the following alright because I can make it log base ten of the number nine log base ten of the number three so you see what I can in other words I can transform this log into a division of two other logs of any base that I want I just have to take the log of this thing and the log of this thing if I want base 17 no problem I can make it log base 17 of nine divided by log base 17 of three so I want log 20 base 24 I can make log base 24 of nine divided by log base 24 of three of course I don't have buttons for all those other weird bases but mathematically they're correct now why do I care about base ten so much alright because basically this means this is the same as log of 9 divided by log of three there is an implied base 10 here because when you don't write it you just assume it's a base 10 now I do have a logarithm base 10 button on my calculator in fact that's why we don't usually on handheld calculators have crazy weird base buttons or menus or anything to be able to take logs of different bases because you don't need to if you need to take a crazy log with a crazy base just transform it to be a log base 10 of this divided by a log base 10 of this so the way you want to read this this transformation of bases this blog can be written as log of this divided by log of this in whatever base that I want usually your is going to convert it to base 10 in order to do the calculations okay let me show you a practical example of why you might want to do that let's say you have log base let's make it base 4 of the number seven and I want you to to simplify that tell me what that's equal to well you all know that if I wanted to I can use the definition of logarithm to try and solve this I can say base to some number equal X on the right-hand side of the equal sign is equal to seven this is what we've been doing for the definition of all logarithms 4 to the power something is equal to 7 that's typically how I've taught you how to solve logarithms you write it as an exponential and then you you try to figure out what this exponent is however 4 & 7 have completely different basses I cannot write them on the left and the right hand side is the same base I can't do it so there's no way that I can solve that thing right however let's just abandon that and let's go down here and say okay instead of evaluating it by using the definition of the logarithm let's just change the base right what this means is that instead of log base four log of the number seven I can write this as the log of this divided by the log of this in any base that I want so I can just make it a base 10 log of the number seven divided by the base 10 log of the number four I have buttons for these on my calculator now of course I could make this a base nine log of seven and divided by base nine log of four or I can make this a base two log divided by base two long all of them are correct you can pick anything you want but base tens the only thing I have a button for so on my calculator I'm gonna go crank in and put log seven and I'll get a decimal back zero point eight four five one this is rounded and truncated of course I'm just carrying four decimals and then zero point six zero to one and so when you when you put that in and do that division you'll get one point four oh four again it says rounded right right so I can say that the log base four base for logarithm of the number seven is equal to this and I can do it in my calculator because of the change of base formula now if I wanted to instead of doing it like this I could also say okay well I'll do you know I could do log base 2 of 7 divided by log base 2 of 4 right if I calculate this it's gonna get exactly the same number or I could say log base 3 of 7 divided by log base 3 of 4 see you're just taking the log of this divided by the log of this in any base you want that's what the change of base formula is it's kind of hard to see the power of that here this is a bunch of letters everywhere you know that's why I kind of write it on the board and then I immediately want to get away from it because when you look at all the letters they all get confusing but when you see real problems you can see how important and how useful it is because of what we're doing here right now remember a long time ago way in the beginning I told you one of the biggest uses of logarithms was to solve exponential equations so how about something like this 3 to the power of 2x is equal to 5 right so this is an exponential equation if you're a human computer and can guess the power of X here needed to make it equal 5 when you raise it then you are much much smarter than me and there's a lot of people that are way smarter than me so good for you but most of us can't calculate like that so how would we solve this we're gonna take log of both sides why am I going to take the log of both sides because I know that a logarithm with a base 3 can cancel this exponential entirely so for instance I can make this log base 3 of 3 to the power 2 X that's what I'm taking the log of and if I do it to the left I also have to do it to the right I can do whatever I want to both sides of the equal sign so since I have a base 3 logarithm and I'm taking the log of a base 3 exponential they cancel all I have left on the left is 2x and I still have this log 3 of the number 5 on the right hand side no problem okay so what I want to do is I want to figure out how to solve this thing but the problem is I have now a base 3 even though I did this and it's like ok cool I've got this I've got a logarithm down here but I don't have a base 3 logarithm button on my calculator so I'm stuck now if I want to write it exactly if I want to write it exactly I could just divide by 2 and say that's the final answer I could just say well you know X is log base 3 of the number 5 divided by 2 this is the exact answer I can of course I can circle that but the problem is what if you want a decimal what if you're trying to calculate the tolerance and all of some part in a design and you only need 3 decimals of accuracy well then you want a number you don't want to leave it as a weird logarithm so what you would do is you would take this step right here and you would go over here and you say well I know that 2x is the log base 3 of the number 5 I don't have a button like this but I know I can change base I can say this is the log base ten of the number five divided by log base ten of the number three again I'm choosing based in here only because I have a button on my calculator that's a base ten I could easily say base for log of 5 base 4 log of 3 base 2 log of 5 base 2 log if they're all correct but none of them are useful so I'm trying to tell you what's basically useful and so then if you want to get the value of x here exactly you could say log base 5 divided by the 2 times the log base base 10 log of 5 base 10 log of 3 now I can circle this and call this exact at least I have base 10 logarithms there but the problem is generally you want numbers so you can say X is equal to what is the base 10 log of 5 0.699 0 of course that's rounded and then here you have 2 times what's the log of 3 is 0.477 1 so when you crank this to a calculator and divide by 2 divided by the point seven seven one point four seven seven one you'll get zero point seven three to six again this is round these are all rounded numbers but does any take logarithms of weird numbers like that you're always gonna get you know infinite does not always but almost always infinitely long trailing decimal so I'm just rounding in here 7 3 2 6 so before knowing the change of base formula you'd basically be stuck right here you would know you could take the log of both sides but you'd be stuck you could solve for x you could get it to this point but you wouldn't know how to actually take the log of this because of the way the numbers I mean you could use the definition of logarithm and say 3 to the power something is 5 but what is that power you still don't know how to solve that because 3 & 5 are weird numbers you can't put them in the same base so this is one of the primary uses of this change a base formula looks a little bit weird in this format but when you crank through it you can see how useful it is in solving equations now what we're gonna do for the rest of the lesson is I'll do a little bit of a mixture of solving some more exponential equations solving some logarithmic equations sometimes we'll need to change the base formula and sometimes we won't and at the end of the lesson I'm gonna derive this change of base formula it's only about five lines of math but I do want you to know where it comes from so it's not a crazy long derivation all right let's say we have logarithm x equals zero point eight five three one eight five three one now when you see logarithm of X and there's no base here you need to assume it's a base ten you need to get used to that when you see don't see anything there it means base ten now how do I solve this well I can use the definition there's a base ten here means 10 to the power of this zero point eight five three one is equal to what's on the left hand side so when you flip it around you can say in calculator ten to the power of whatever this decimal is you'll get seven point one three again this is rounded this is the value of x these are a little kind of short simple little problems just to show you that when you have decimals you can put them in your calculator raise it the power of ten you can take logarithms with decimals and think it's kind of trying to get you to start using your calculator because sometimes you will almost certainly need to use a calculator to finish the problem off what if you have logarithm of the number X is equal to negative 1.8 now again there's no base here so that means you need to assume it's a base ten so the way you solve this you should say base ten to the power of negative 1.8 is equal to what's on the left hand side X so then you go to your calculator stick a negative 1.8 raised 10 to the power of that there's a should be a button on your scientific calculator to take ten and raise it to the power of something and you would get zero point zero one five eight again this is rounded zero one five eight all right you just have two more of these small problems and then we're going to derive the change of base formula what about three to the power of X is thirty now this is an exponential equation how would you solve it well in order to get rid of the exponential here and get this X off by itself I need to get rid of the exponential so I'm going to take the logarithm of both sides but if I'm going to take the log of both sides and kill this it needs to be a base three logarithm three to the power of X base three logarithm on the right of the number 30 base 3 log on both sides so when you actually do that the logarithm and the exponential cancel just leaving you with the variable x equals log base 3 of 30 now up until now you would just circle this as your answer a log base 3 of 30 that's totally fine however what if you want a decimal how would you calculate it in a calculator we're going to use the change of base formula so what you'll get is you'll say well this is a logarithm base 10 log of this 30 divided by the base 10 logarithm of the base 3 again I can use any base I want here I can use base 7 base 10 base 14 base 20 s it doesn't matter but base 10 is the only one I'm gonna use because that's the button on my calculator I have so the base 10 log of 30 comes out to be 1 point 4 7 7 this is rounded okay all right and yeah I guess I'll do it like this and then on the bottom you'll have a log of 3 0.477 and so then what you'll find is X is equal to when you take those numbers and divide you'll get 3 points 0 9 6 of course this is approximate again because this is an approximation this number is exact so if you're asked to tell me the exact value of x this is what I would write down now you could leave it like this sure but generally you want to convert to base 10 so that somebody else could finish the calculation and use the change of base formula to get there this is the exact value this is acceptable also but in base 10 this is the exact value and then of course you get the numerical approximation to it all right now what we want to do is we want to show you where things come from I want to show you where this change of base formula comes from it's not a hard derivation it only takes a few seconds but then you'll feel comfortable knowing where everything comes from so what I want to do first is I want to make a definition I'm gonna say let the following thing be true logarithm base a of some number X is equal to Y I can in a proof I can let anything equal in my initial step I can let anything equal I'm just assigning letters basically I'm saying a base a log of some number X is equal to some number y of course that can make any assignment I want but if I make that that Association then you have some consequences of that if I make that Association then I say that a to the power of Y is X how do I know that because the base to the power of this is equal to this that's just the definition of the logarithm then I can say I can do whatever I want to this because if this is true then this is true so then the next thing I want to do is I want to try to take the logarithm of both sides right and in order to keep everything general I'm gonna take some logarithm of both sides but I'm gonna take it with a different base B here's a to the Y power log base B of X now you might say well how do you know to do that well the truth is this is a proof nobody here watching this I expect to really know how to prove this ahead of time you probably see this thing one time and say oh I get it okay so you're not gonna really know ahead of time that this is gonna work but I'll tell you the reason I'm taking a log with some other base B is because I know that where I'm trying to get to has a logarithm with a base B over here and then I'm starting with a logarithm of some other other different base a so the only way I'm gonna really be able to get to proving that thing is to have introduced all base B logarithm somewhere and I know it's legal to take the logarithm of both sides of an equation anytime I want so I'm gonna do that now here you can see you have a logarithm here but this exponent can come out with the law of exponents so Y times log base B of whatever is left over a and this is log base B of X all I've done is using the law of logarithms here but here's the thing I know what Y is equal to Y is equal by definition to this line up here so I can substitute that in the log base a of the number X multiplied by the log base B of the number a is equal to log base B of the number X right and so basically believe it or not we're actually home free let me just solve for this term log base a of some number X is equal to log base B of some number X divided by log base B of some number a this is what I've proved I basically said it's the logarithm with some base a of some number X is equal to the log of X divided by the log of the base in any other base I want this right here is exactly what I write on the board here right so we've proved that guy now do I expect you to care about that do I expect you to to write that you know down every night and and and and cuddle up to it and love it no of course not but I want you to know where things come from I want you to know that they just don't come down from the sky and I also want you to know that something is weird-looking as this formula is not that hard to understand I mean yes if you don't know what a log is it looks hard but I'm just it's a very logical set of steps I'm letting this equal to this this is the definition of the logarithm I take the log of both sides I use the law logarithms to pull it out and then from then I just rearrange and figure out this is true this is very useful you would use it in algebra you would use it in trig you would use it in calculus extensively it's called the change of base formula so make sure you understand how to solve all of these problems follow me on to the next lesson we'll get some more practice with solving more exponential equations and using the change of base formula of logarithms